# What is the $C_p/C_v$ ratio?

Recently in our chemistry class we read that the $C_p/C_v$ ratio of the inert gases is 1.66 thus they show inert nature. I asked my teacher what was $C_p/C_v$ ratio but he didn't answer me. So I want to know what the $C_p/C_v$ ratio really is?

• Is there something on the wikipedia page that you don't understand? en.wikipedia.org/wiki/Heat_capacity_ratio Commented Jan 24, 2014 at 2:39
• I think the more interesting question to ask is why a value of $\gamma = 5/3$ indicates a gas is inert... but that should be posted as a separate question! Commented Jan 24, 2014 at 2:41

The ratio is called the heat capacity ratio; $C_P$ expresses the heat capacity (the heat energy required to change a temperature by a certain amount) of a gas held at a constant pressure, $C_V$ the heat capacity of a gas held at constant volume.
I personally like to view the ratio (which I usually call the adiabatic index) as a measure of the degrees of freedom of the gas: $$\gamma=1+\frac{2}{f}\longleftrightarrow f=\frac{2}{\gamma-1}$$ where $C_P/C_V\equiv \gamma$. A monatomic gas has $f=3\to \gamma=1+2/3=5/3=1.\overline{66}$ (the 3 comes from translational motions, rather than vibrational or rotational motions that can occur for diatomic molecules in which $f=5\to\gamma=1+2/5=7/5=1.4$).